That's exactly what I had to do to our bathroom, and it took ages. Jamie replaced the silicon and I got a grout whitener in a tube. I've used the pen before and found it took longer and wore off faster! I didn't want to spend a ton of money on a new bathroom when mine's perfectly functional, so a tidy up, a paint and a new wooden benchtop and handles made it look pretty nice. Now I just really covet some warm brushed nickle tapware I can't afford...